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Full text of "Solution of a system of differential equations occurring in the theory of radioactive transformations"

\jP Biodiversity fe^HeriUge http://www.biodiversitylibrary.org/ Proceedings of the Cambridge Philosophical Society, Mathematical and physical sciences. Cambridge [etc.]Cambridge Philosophical Society. http://www.biodiversitylibrary.org/bibliography/44060 V. 15 (1908-10): http://www.biodiversitylibrary.org/item/97262 Page(s): Page 423, Page 424, Page 425, Page 426, Page 427 Contributed by: Smithsonian Institution Libraries Sponsored by: Smithsonian Generated 27 September 201 1 1 :51 PM http://www.biodiversitylibrary.org/pdf3/008070400097262 This page intentionally left blank. Mr BatemaUy Solution of a s I equations, etc. 423 The solution of a system of differential occurring in the theory of radio-active transformations. By H. Bateman, M,A., Trinity College. [Read 21 February 1910.] 1. It has been shown by Prof. Rutherford * that the amounts of the primary substance and the different products in a given • . rh ^« *■ ii T*ij1 i r* quantity of radio-active matter vary according to the system of d equations, dt d^ dt \P \iJr — A»2y -— = \^Q — Xjit (1), d^ dt A? a — \a1 » • ; where P, Q, R,S,T,... denote the number of atoms of the primary substance and successive products which are present at time t. Prof Rutherford has worked out the various cases in which there are only two products in addition to the primary substance, and it looks at first sight as if the results may be extended to any number of products without much labour. Unfortunately the straightforward method is unsymmetrical and laborious, and as the results of the calculations are needed in some of the researches which are being carried on in radio-activity the author has thought it worth while to publish a simple and symmetrical method of obtaining the required formulae. 2. Let us introduce a set of auxiliary, quantities p (^), q (a?), r {a))y depending on a variable ^ and connected with the quantities P{t)y Q(t), R{t\ .••by the equations, p{x) 00 00 e-''^P(t)dt, q{x) e~''^Q{t)dt (2). It is easily seen that 00 e 3/5 dP dt dt A 00 P(0) + ^ e-^^Pif)dt (3) JO ? Pis-^xp J * Radio-activity, 2nd edition, p. 331. Ml' Bateman, Solution of a system of differential equations where p is written for p (w), and P^ for P (0), the initial value of P (t). Multiplying equations (1) by e~^\ and integrating from to 00 with regard to t we obtain the system of equations XP X^Q Qo Sn a-q />*^'* -\p \ip — X^q ^2q A4.9 from which the values of jj, q, r may be obtained at once. If Q.^R S^ m m present initially, we have 0; i.e. if there is only one substance P X + Xi <1 {x + Xi) {x + Xo) ' /v» Aj A2 JTq {x + Xj) (^ + X2) {x + X3) ' and for the ni\\ product V {x) Ax Ag • • • A^_i X"^, (^ + Xi) (^ + Xo) . . . {x+ Xn) * Putting this into partial fractions, we have v{x) a + a ^^t^^^ T I. ^ + Xi ^ + X 1 • • • c n X + Ayj where c Aj A2 • • • Ajj 2 -i (X Xl) (X; Xl) . . . (X;i — Xj) Go /vj /V^ • • • "^ji 1 -^ y \\i — X2) (Xg — Xo) ... (X il Xo) (6). • • « * 6 uC, To obtain the corresponding function N{t) we must solve the integral equation V (x) 00 e-^^ N {t) dt Now it has been shown by Lerch^ that there is only one continuous function N (t) which will yield a given function v (x) ; hence if we can find a function which satisfies this condition it will be the solution of our problem. It is clear, however, that 1 ^ + X 00 ^-xt ^ ^-Kt ^l . hence the above value of v {x) is obtained by taking N {t) ^ c^e'^^^^ ^^ c^e^^^^ + ... Cne'^^'^ e the constants have the values given by (6). (7), * Acta Mathematical 1903, p. 339. occurring in the theory of radio-active transformations. In the case when Q (0), R (0), . . . are not zero, we have P •* iC+Xi' ? \,P + Qo {X + A-i) (il? + Xa) ^ + Xo r ^1^2-^ + A2V0 + -Bo •(8), (x + \i) (a; + Xa) («^ + 5^3) (« + ^2) («? + ^3) (« + ^3) etc. and we may obtain the values of P, Q, R by expressing these quantities in partial fractions as before. The complete solution for the case of a primary substance P and three products Q, R, S is P = Poe-^'^ Q \ X \ Png-^'^ + ^''^'- + Qo) e-'^* R A,^ A^2 -^0 (Xo — Xi) (X,3 — Xj) 6 Aif + Xi — X A.2 A.2 -to + X2 Vo (Xj — X2) (X3 — X2) X A^9 e At^iyjj + \l\2Jr + X2Q (Xj — X3) (Xg — X3) X.2 — X + R e Aqt 8 KiK^*^^-^ Q (Xo X 1 ) (X; \i) (X. Xi) e AjT + * Xi Xg X3 X -. + X'2 X3 t^Q (Xi — X2) (Xo — X2) (X4 — Xo) (X3 — X2) (X4 — Xy)^ e \of + XiX2X3-r m'^^ t ■ 1 I + A-2 A3 (^Q _(Xi — X3) (X2 — X3) (X4 — X3) (X2 — X3) (X "^3) + ^4 — ^1 Q-\st + Xt Xo Xct ± X tkj MJ _(Xl — X4) (X2 — X4) (X; \) + Xo X3 {^Q (X2 — X4) (X3 — X4) X3 — X4 J e x,t The solution may evidently be obtained by superposing the solutions of the cases in which the initial values of P, Q, R, S are given by (1) P(0) = Po, (2) P(0) = 0, (3) P(0) = 0, (4) P(0) = 0. 0, R(0) 0, S{0) 0. QiO) Q{0) Q(0) = 0, R(0) = 0, 5(0) = ^0. Q„ R{0) = 0, S{0) = 0. 0, R(0) = Ro, S{0) = 0. The method is perfectly general, and the corresponding formulae for the case of ?i— 1 products may be written down at once by using (6). The general formula covers all the four cases {Radio-activity, pp. 331 — 337). For instance in Case 2 when initially there is radio-active equilibrium, we have n X,P Xov^o — K^l\,(^ — A^>O0, 426 Mr Bateman, Solution of a system of differential equations The solutions are then P n \ e \,t Q n A- \ e ^i^ -f- n^Xj -^^f^m R Xo7? 2"0 (Xo — Xj) (X. K) i Xo (X^ — Xo) Xi?lo e Agf : e Kot + {\l — X2) (X3 — Xo) Xj X2 "Hq ^^ (^1 **" Xo) (X2 •— X3) e Agf S Xo Xo 71 (X2 — Xi) (Xo — Xi) (X4 — Xj) e Kt + Xi Xo V A O + (Xi — X2) (X3 — X2) (X4 — Xg) e AciV (Xi — Xg) (X2 — X3) (X. ^) e Aqt + XiX2X3?l (Xi — X4) (Xj X4) (Xg X4) e' •^4,* The solution for the ease of ?i— 1 products is given by i\^ = 2a Ov w Ai»t ^j where the constants Cr are obtained by expressing 1 2 3 • • • A/ ?i— 1 00 (00 -\- Xi) (^' + Xg) ... (^ + X^i) in partial fractions. The method by which the solution of the system of differen- tial equations has been obtained is really of very wide application and may be employed to solve problems depending on a partial differential equation of the form f(1 1 i provided the initial value of V is known. dV dt V, For if we put QO u(s)= I e-'^ V{t)dt, Jo su (s) — Vq 00 dV e'^'^-^dt, ot it appears that it(s) satisfies the partial differential equation f( d d d \ \da) ' dy' dz^ U'\-su+ Fo = (9). Further, if V satisfies some linear boundary condition which is independent of t the function ii will generally satisfy the same boundary condition. This function {%i) must be obtained from the occurring in the theory of radio-active transformation^. 427' differential equation (10) which is simpler than (9), inasmuch as it depends upon fewer independent variables. Tn many cases the solution of the integral equation ^t (s) 00 e-'^ V{t) dt may be calculated by means of the inversion formula * V{t) 1 27ri e*^ u (?) d^, e where c is a contour which starts at — oo at a point below the real (r) returns to - x at a point above the real axis, as in the figure. The conditions to be satisfied by ii (^) that this inversion formula may be applicable have not yet been expressed in a concise form. The formula may be used to obtain the solution of a problem in the conduction of heat when we require a solution of d'V dV dx^ dt ' t • « which satisfies the boundary conditions V=^ when ^ = and a) = a^ V =f(,x) when i = 0, The solution found in this way is identical with the one given in Carslaw's Fourier s Series and Integrals, p. 383, * A particular case of this formula has been given by Pincherle, Bologna Memoirs, 10 (8), 1887.